CSCI 5444: Introduction to the Theory of Computation


Relevant Textbooks


Course Objectives

The objective of this course is provide an introduction to the theory of computation covering the following three branches of theoretical computer science:
  1. Automata Theory
    • Formalization of the notion of problems via formal languages
    • Formalization of the notion of computation using "abstract computing devices" called automata
    • Understanding a hierarchy of classes of problems or formal languages (regular, context-free, context-sensitive, decidable, and undecidable)
    • Understanding a hierarchy of classes of automata (finite automata, pushdown automata, and Turing machines)
  2. Computability Theory
    • Understanding Church-Turing thesis (Turing machines as a notion of "general-purpose computers")
    • Understanding the concept of undecidability , i.e., when a problem can not be solved using computers
    • How to show undecidability using the concept of problem reduction
  3. Complexity Theory
    • Complexity classes : how to classify decidable problems based on their time and space requirements
    • Complexity classes P and NP, and Intractability (NP-completeness)
    • How to prove NP-completeness?
    • Space Complexity: NL-completeness and PSAPCE-completeness

Topics Covered

  1. Regular Languages (3 weeks)
    • Deterministic finite-state machines
    • Nondeterministic finite-state machines
    • Regular expressions
    • Properties of regular languages
    • Languages that aren't regular: pumping lemma
  2. Context-Free Languages (2 weeks)
    • Context-free grammars
    • Pushdown automata
    • Properties of Context-free languages
    • Languages that aren't context-free: pumping lemma for CFLs
  3. Computability Theory (4 weeks)
    • Turing machines and their variants
    • Church-Turing thesis
    • Decidable languages
    • Undecidability
    • Proving Undecidability of a given problem using problem reductions
    • Rice's theorem
    • Famous undecidable problems such as Post Correspondence Problem (PCP), Tiling problem, halting problems for multistack and two-counter machines.
  4. Complexity Theory (3-4 weeks)
    • Time and space complexity
    • Complexity classes P and NP, and NP-Completeness
    • Famous NP-complete problems
    • Complexity class PSPACE and Pspace-Completeness
    • Complexity classes L and NL, and NL-completeness
  5. Special Topics (guest lectures and class projects: presentations in Week 16)
    • Monadic Second-Order Logic and Automata (Elements of Finite Model Theory by Leonid Libkin)
    • Regular transformations on words and trees (TBA)
    • Descriptive complexity (Descriptive Complexity by Neil Immerman)
    • Randomized Computation (Computational Complexity by Sanjeev Arora and Boaz Barak)
    • Quantum Computation (Computational Complexity by Sanjeev Arora and Boaz Barak)
    • Interactive proofs and complexity class IP (Computational Complexity by Sanjeev Arora and Boaz Barak)
    • PCP Theorem and hardness of Approximation (Computational Complexity by Sanjeev Arora and Boaz Barak)
    • Timed and hybrid Automata (TBA)
    • Probabilistic Automata (TBA)


The overall grade will be based on a cumulative score computed by adding together the grades from:

Schedule and Lecture Notes

# Date Description Chapter
1 August 25 Introduction to theory of computation 0

Part One: Automata Theory

2 Week 1 — August 27 Regular languages and Deterministic Finite Automata 1.1
3 Week 2 — September 1 Nondeterministic Finite Automata (Subset Construction and Alternation) 1.2
4 Week 2 — September 3 Closure Properties for Regular Languages 1.1
5 Week 3 — September 8 Regular Expressions 1.3
6 Week 3 — September 10 Non-Regular languages: Pumping Lemma 1.4
7 Week 4 — September 15 Logic and Regular Languages lecture notes
8 Week 4 — September 17 Context-Free Languages: Grammars and Derivations 2.1
9 Week 5 — September 22 Pushdown Automata 2.2
10 Week 5 — September 24 Non-Context-Free Languages 2.3
11 Week 6 — September 29 Closure properties of CFLs
12 Week 6 — October 1 Wrap-up of Regular Languages and CFLs 2.1 — 2.3
13 Week 7 — October 6 In-Class Quiz I 1 and 2

Part Two: Computability Theory

14 Week 7 — October 8 Turing machines 3.1
15 Week 8 — October 13 Variants of Turing machines 3.2 and 3.3
16 Week 8 — October 15 Decidability: Decidable Languages 4.1
17 Week 9 — October 20 Halting Problem: Diagonalization and Reductions 4.2
18 Week 9 — October 22 Reductions: More undecidable problems 5.1, 5.2
19 Week 10 — October 27 Logics and Decidability 6.2
20 Week 10 — October 29 Wrap-up: Turing machines and decidability 3-4-5-6
22 Week 11 — November 3 In-class Quiz II 3-4-5-6

Part Three: Complexity Theory

23 Week 11 — November 5 Complexity 7.1 and 7.2
24 Week 12 — November 10 NP, co-NP, polynomial-time reductions and NP-completeness 7.3
25 Week 12 — November 12 NP-complete problems and reductions 7.4
26 Week 13 — November 17 Space Complexity Classes: Savitch's theorem
27 Week 13 — November 19 PSPACE and PSPACE-complete problems 7
28 Week 14 — November 24 In-class Quiz III
29 Week 15 — December 01 Special Topics: TBA
30 Week 15 — December 03 Special Topics: TBA


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